[#P2988] The limit of the minimum overlap quotient

Problem. Erdős Problem 36: For each positive integer $n$, consider all partitions of $\{1, 2, \dots, 2n\}$ into two disjoint $n$-element subsets $A$ and $B$. For integers $a \in A$ and $b \in B$, define the overlap of $A$ and $B$ at difference $k \in \mathbb{Z}$ to be the number of pairs $(a,b)$ with $a - b = k$. Define the maximum overlap $M(A,B)$ to be the largest overlap over all integer differences $k$. Define $M(n)$ to be the minimum of $M(A,B)$ over all such partitions $(A,B)$. Define the minimum overlap quotient by \[Q(n) = \frac{M(n)}{n}.\] Does the limit $\displaystyle\lim_{n \to \infty} Q(n)$ exist, and if so, what is its exact value?

A finite mathematical diagram showing a two-color partition and counts of cross-differences. — A two-color partition with cross-difference counts.

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1Context

This is Erdős Problem 36, also known as the minimum overlap problem. The problem concerns the asymptotic behavior of the minimum possible maximum overlap when splitting the first $2n$ integers into two equal parts. The limit's existence has been established, with the best known bounds being approximately $0.379005 < \liminf Q(n) \leq \limsup Q(n) \leq 0.3809268534330870$, but the exact value remains unknown.

2Definitions

Definition 1 (For finite sets $A, B \subset \mathbb{Z}$ and $k \in \mathbb{Z}$, the overlap $\operatorname{Overlap}(A,B,k)$). For finite sets $A, B \subset \mathbb{Z}$ and $k \in \mathbb{Z}$, the overlap $\operatorname{Overlap}(A,B,k)$ is the number of ordered pairs $(a,b) \in A \times B$ such that $a - b = k$.

Definition 2 (The maximum overlap $M(A,B)$). The maximum overlap $M(A,B)$ is $\sup_{k \in \mathbb{Z}} \operatorname{Overlap}(A,B,k)$, which equals the maximum since only finitely many differences occur.

Definition 3 (For $n \geq 1$, $M(n)$). For $n \geq 1$, $M(n)$ is the minimum value of $M(A,B)$ over all partitions of $\{1,2,\dots,2n\}$ into two disjoint $n$-element subsets $A$ and $B$.

Definition 4 (The minimum overlap quotient). The minimum overlap quotient is $Q(n) = M(n)/n$.

3What counts as a solution

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 36: For each positive integer $n$, consider all partitions of $\{1, 2, \dots, 2n\}$ into two disjoint $n$-element subsets $A$ and $B$. For integers $a \in A$ and $b \in B$, define the overlap of $A$ and $B$ at difference $k \in \mathbb{Z}$ to be the number of pairs $(a,b)$ with $a - b = k$. Define the maximum overlap $M(A,B)$ to be the largest overlap over all integer differences $k$. Define $M(n)$ to be the minimum of $M(A,B)$ over all such partitions $(A,B)$. Define the minimum overlap quotient by \[Q(n) = \frac{M(n)}{n}.\] Does the limit $\displaystyle\lim_{n \to \infty} Q(n)$ exist, and if so, what is its exact value?

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Current status and unresolved remainder). OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 36 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 36: For each positive integer $n$, consider all partitions of $\{1, 2, \dots, 2n\}$ into two disjoint $n$-element subsets $A$ and $B$. For integers $a \in A$ and $b \in B$, define the overlap of $A$ and $B$ at difference $k \in \mathbb{Z}$ to be the number of pairs $(a,b)$ with $a - b = k$. Define the maximum overlap $M(A,B)$ to be the largest overlap over all integer differences $k$. Define $M(n)$ to be the minimum of $M(A,B)$ over all such partitions $(A,B)$. Define the minimum overlap quotient by \[Q(n) = \frac{M(n)}{n}.\] Does the limit $\displaystyle\lim_{n \to \infty} Q(n)$ exist, and if so, what is its exact value?^[1]

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Record	Kind	Assessment
OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 36 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 36: For each positive integer $n$, consider all partitions of $\{1, 2, \dots, 2n\}$ into two disjoint $n$-element subsets $A$ and $B$. For integers $a \in A$ and $b \in B$, define the overlap of $A$ and $B$ at difference $k \in \mathbb{Z}$ to be the number of pairs $(a,b)$ with $a - b = k$. Define the maximum overlap $M(A,B)$ to be the largest overlap over all integer differences $k$. Define $M(n)$ to be the minimum of $M(A,B)$ over all such partitions $(A,B)$. Define the minimum overlap quotient by \[Q(n) = \frac{M(n)}{n}.\] Does the limit $\displaystyle\lim_{n \to \infty} Q(n)$ exist, and if so, what is its exact value?	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 36 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 36: For each positive integer $n$, consider all partitions of $\{1, 2, \dots, 2n\}$ into two disjoint $n$-element subsets $A$ and $B$. For integers $a \in A$ and $b \in B$, define the overlap of $A$ and $B$ at difference $k \in \mathbb{Z}$ to be the number of pairs $(a,b)$ with $a - b = k$. Define the maximum overlap $M(A,B)$ to be the largest overlap over all integer differences $k$. Define $M(n)$ to be the minimum of $M(A,B)$ over all such partitions $(A,B)$. Define the minimum overlap quotient by \[Q(n) = \frac{M(n)}{n}.\] Does the limit $\displaystyle\lim_{n \to \infty} Q(n)$ exist, and if so, what is its exact value?^[1] Relevance to this problem For erdos problem 36, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 36 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 36: For. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 36 as open. The unresolved remainder is the full displayed statement. A complete resolution must satisfy this condition: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 36: For each positive integer $n$, consider all partitions of $\{1, 2, \dots, 2n\}$ into two disjoint $n$-element subsets $A$ and $B$. For integers $a \in A$ and $b \in B$, define the overlap of $A$ and $B$ at difference $k \in \mathbb{Z}$ to be the number of pairs $(a,b)$ with $a - b = k$. Define the maximum overlap $M(A,B)$ to be the largest overlap over all integer differences $k$. Define $M(n)$ to be the minimum of $M(A,B)$ over all such partitions $(A,B)$. Define the minimum overlap quotient by \[Q(n) = \frac{M(n)}{n}.\] Does the limit $\displaystyle\lim_{n \to \infty} Q(n)$ exist, and if so, what is its exact value? Open full record →
Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 36: For each positive integer $n$, consider all partitions of $\{1, 2, \dots, 2n\}$ into two disjoint $n$-element subsets $A$ and $B$. For integers $a \in A$ and $b \in B$, define the overlap of $A$ and $B$ at difference $k \in \mathbb{Z}$ to be the number of pairs $(a,b)$ with $a - b = k$. Define the maximum overlap $M(A,B)$ to be the largest overlap over all integer differences $k$. Define $M(n)$ to be the minimum of $M(A,B)$ over all such partitions $(A,B)$. Define the minimum overlap quotient by \[Q(n) = \frac{M(n)}{n}.\] Does the limit $\displaystyle\lim_{n \to \infty} Q(n)$ exist, and if so, what is its exact value?	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 36: For each positive integer $n$, consider all partitions of $\{1, 2, \dots, 2n\}$ into two disjoint $n$-element subsets $A$ and $B$. For integers $a \in A$ and $b \in B$, define the overlap of $A$ and $B$ at difference $k \in \mathbb{Z}$ to be the number of pairs $(a,b)$ with $a - b = k$. Define the maximum overlap $M(A,B)$ to be the largest overlap over all integer differences $k$. Define $M(n)$ to be the minimum of $M(A,B)$ over all such partitions $(A,B)$. Define the minimum overlap quotient by \[Q(n) = \frac{M(n)}{n}.\] Does the limit $\displaystyle\lim_{n \to \infty} Q(n)$ exist, and if so, what is its exact value?^[1] Relevance to this problem For The limit of the minimum overlap quotient, record erdos-problem-36-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 36: For each positive integer $n$, consider all partitions of $\{1, 2, \dots, 2n\}$ into two disjoint $n$-element subsets $A$ and $B$. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 36: For each positive integer $n$, consider all partitions of $\{1, 2, \dots, 2n\}$ into two disjoint $n$-element subsets $A$ and $B$. For integers $a \in A$ and $b \in B$, define the overlap of $A$ and $B$ at difference $k \in \mathbb{Z}$ to be the number of pairs $(a,b)$ with $a - b = k$. Define the maximum overlap $M(A,B)$ to be the largest overlap over all integer differences $k$. Define $M(n)$ to be the minimum of $M(A,B)$ over all such partitions $(A,B)$. Define the minimum overlap quotient by \[Q(n) = \frac{M(n)}{n}.\] Does the limit $\displaystyle\lim_{n \to \infty} Q(n)$ exist, and if so, what is its exact value? Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 36 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 36 was open at commit 8138974387d9030542daabe67faaa33eff9356f8.
The pinned Formal Conjectures declaration was matched by problem number and source locator.
The controlled TheoremDB source corpus was checked for an already published record with the same slug.

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The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemThe limit of the minimum overlap quotient

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Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

Constructing a union from non-cancelling intersectionscombinatorics
Plane tilings by every five-cell lattice animalcombinatorics
Cycle Double Cover Conjecturecombinatorics

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“The limit of the minimum overlap quotient.” TheoremDB. P2988. Problem statement; statement text SHA-256 b5b0f1459bcfea62278dba37661c5839aa008e3f1421c9ab94feb9e99efb2d39. https://theoremdb.org/statement/?ref=P2988

BibTeX

@misc{theoremdb-problem-b5b0f1459bcfea62278dba37661c5839aa008e3f1421c9ab94feb9e99efb2d39,
  title = {{The limit of the minimum overlap quotient}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 b5b0f1459bcfea62278dba37661c5839aa008e3f1421c9ab94feb9e99efb2d39},
  url = {https://theoremdb.org/statement/?ref=P2988}
}

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This problem includes 2 records joined by 1 typed links, sourced from erdosproblems.com^[1], current as of July 31, 2026.

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1References

Packet source. Erdős Problems database, Problem 36, maintained status record. Erdős Problems record 36, checked 2026-08-01. Problem 36; status field and linked bibliography. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Records the current open status and links the literature attached to this exact numbered problem.Also cited at Problem 36; status snapshot 8138974387d9030542daabe67faaa33eff9356f8.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For The limit of the minimum overlap quotient: Records the current open status and links the literature attached to this exact numbered problem.Source named by the research packet.
Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 36 entry in data/problems.yaml. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Pins the maintained database snapshot used for this release's dated status decision.Source used to assess the problem's recorded status.For The limit of the minimum overlap quotient: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 36. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/36.lean:L257; theorem erdos_36; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. ↗reference database · primary source · commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 · checked 2026-07-31Source use: original summary.Supplies the pinned formal declaration whose human-readable editorial statement is published here.Source used to assess the problem's recorded status.For The limit of the minimum overlap quotient: Supplies the pinned formal declaration whose human-readable editorial statement is published here.

Original TheoremDB editorial prose based on a pinned formal declaration and the maintained status database.

[#P2988] The limit of the minimum overlap quotient

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